Problem: Zane is a dangerous fellow who likes to go rock climbing in active volcanoes. One time while inside a volcano, he heard some rumbling, so he decided to climb up out of there as quickly as he could. He climbed up at a rate of $4$ meters per second. After $3$ seconds, he was $13$ meters below the edge of the volcano. How far was Zane below the edge of the volcano when he started climbing?
Zane climbed up at a rate of $4$ meters per second, so he climbed up $4T$ meters in $T$ seconds. The remaining distance Zane had to climb is found by taking the original distance and subtracting from it the distance Zane had already climbed. We can express this with the equation $R=D-4T$, where: $R$ represents Zane's remaining distance to climb (in meters) $D$ represents Zane's original distance from the edge of the volcano (in meters) $T$ represents the time (in seconds) We want to find $D$, so let's first solve the equation for $D$ : $ \begin{aligned}R&=D-4T\\ D&=R+4T\end{aligned}$ Now, we know that after $3$ seconds $(T={3})$, Zane was $13$ meters below the edge of the volcano $(R={13})$. Let's plug these values into the equation to find the value of $D$. $ D={13}+4\cdot{3}=25$ Therefore, Zane was $25$ meters below the edge of the volcano when he started climbing. To find how long it took Zane to reach the edge of the volcano, we can plug $R=0$ into the equation and solve for $T$. $ \begin{aligned}25&=0+4T\\ 4T&=25\\ T&=6.25\end{aligned}$ Zane was $25$ meters below the edge of the volcano when he started climbing. It took Zane $6.25$ seconds to reach the edge of the volcano.